MHD Studies For Double Tearing Modes In Tokamak Plasmas

Tearing mode is a dominant magnetohydrodynamic (MHD) instability in tokamakplasmas. One of the proposed improvements for high performance magnetic confinement inadvanced tokamaks is the configuration with a reversed magnetic shear. On the other handhowever this type of magnetic configuration will induce double tearing modes (DTMs). Theislands at both resonance surfaces with the same safety factor will drive each other violently,resulting in severe destructions to magnetic confinement in tokamak discharges. In the lasttwo decades, theoretical and numerical investigations have focused on the linear and earlynonlinear developments of DTMs. Owing to its complexity, the late nonlinear evolution ofDTMs, in particular the magnetic topology change and relevant dynamic growth, are still farfrom well understood.In this thesis, we adopt the reduced resistive MHD model to study the entire nonlinearevolution of DTMs with emphasis on resistivity-scaling laws of the growth rate in variousregimes. The problem of magnetic topology evolution and related dynamic processes in theentire nonlinear development of DTMs is solved for the first time.In Chapterâ… , a review on tearing modes in space and laboratory plasmas is introduced,including the basis of linear and nonlinear ordinary tearing mode as well as DTMs.In Chapterâ…¡, the reduced resistive MHD model for compressible plasmas is employedto investigate the entire nonlinear evolution of DTMs. Resistivity dependences for variousphases are studied and shown by scaling analysis for the first time, improving theunderstanding for the nonlinear fast growth phase in previous articles [Z. Chang et al., Phys.Rev. Lett. 77, 3553 (1996); Y. Ishii et al., Phys. Rev. Lett. 89, 205002 (2002)]. It is found thatin the nonlinear evolution of DTMs, the first and second phases are non-constant-ψSweet-Parker regime with a～η^1/2 scaling and Rutherford regime with a～η¹ scaling,respectively. The third phase is the Wang-Bhattacharjee-Ma fast reconnection regime with a～η^1/5 scaling due to the merging of magnetic separatrices which may be regarded as asteadily inward flux driven. Finally, an oscillating decay phase follows. The magneticreconnection between islands and equilibrium field lines then starts when the magnetic fieldlines in the region between the rational surfaces are fully reconnected. As a result, the newlygenerated field lines push the islands away, explaining the phenomena of the radial positionexchange of the islands in the late nonlinear evolution of DTMs. In addition, a prediction forthe final stage of magnetic reconnection in multiple resonance surface systems is brought out. In Chapterâ…¢, the evolution of the plasma velocity in the nonlinear development ofDTMs is investigated. The plasma velocity is limited inside of each island when the islandwidth is small in the early phases. In the third phase however, the island width is large enoughfor flows to couple together and form big vortices, and simultaneously the poloidal shearedflow layers also form in the region close to the tips of the magnetic islands. Furthermore, thevortices and the flow layers still remain for a while after the merging of magnetic separatrices.In the final stage of magnetic reconnection, the plasma flows develop along magnetic fieldlines with small velocity and form several narrow vortices.In Chapterâ…£, the effects of the distance between the resonance surfaces and the shearstrength of the initial magnetic field on the nonlinear evolution of DTMs are studiednumerically. It is found that the mode is most unstable if the distance between the resonancesurfaces is in a range of～0.3L₀, where L₀ is the typical length for the poloidal variation ofthe magnetic field. Moreover, increasing the initial shear strength will effectively intensify theMHD activities, including the evolution time, and maximums of the kinetic energy, poloidalvelocity, poloidal velocity shear, and current. These influences are explained in the aspect ofmagnetic energy release. An analysis for the numerical results in this chapter could cast lightson the magnetic configuration design in tokamaks.As a specific example, a quasi-linear theory of incompressible MHD model is used inChapterâ…¤to study the effects of the m=0 harmonics on the early quasi-linear stage ofm=1 double tearing modes. In the early growth, it is found that m=1 harmonicsdominates the linear growth and the effect of the m=0 harmonics can be neglected. As themode grows nonlinearly however, the contribution of the m=0 harmonics begins to exceedthat of the m=1 harmonics and dominates the mode. It is the coupling between the m=0and m=1 harmonics that makes the total current pinched around the resonance surfaces toform sharp and narrow profiles. As a result, the equilibrium poloidal magnetic field isdistorted to reduce the destabilizing free energy stored in the anti-parallel magnetic field. Them=0 harmonics plays a dominant role in the final stage of magnetic reconnection. In thecase without the m=0 harmonics, the m=1 harmonics will make the mode grow linearlyfor a long period. The profiles of the current, magnetic field, and plasma velocity in bothcases are compared.Finally, a brief summary ends the thesis.